The dual fibration in elementary terms 5 1 0 Anders Kock 2 n a J We give an elementary construction of the dual fibration of a fibration. It 8 does not use the non-elementary notion of (pseudo-) functor into the category of ] categories. In fact, it is clear that the construction we present makes sense for T C internalcategoriesand fibrationsin anyexact category. . ThedualfibrationofafibrationX →BoverBisdescribedine.g.[Borceux] h t II.8.3 via a pseudofunctor F : Bop →Cat (the category of categories), by com- a m posingF withthe(covarariant!) dualizationfunctorCat →Cat;choosingsuchan [ F is tantamount to choosing a cleavage for the fibration. In the present section, we give an alternative description of the dual fibration, which is elementary and 1 v choice-free. 7 4 9 1 Fibrations 1 0 . 1 Werecall here someclassicalnotions. 0 Letp :X →B beanyfunctor. Fora :A→BinB,andforobjectsX,Y ∈X 5 1 withp (X)=Aandp (Y)=B,lethoma (X,Y)bethesetofarrowsh:X →Y inX v: with p (h)=a . For any arrow x :C→A, and any object Z ∈X with p (Z)=C, i X post-compositionwithhdefines a map r a h∗ :homx (Z,X)→homx .a (Z,Y). (we compose from left to right). Recall that h is called Cartesian if this map is a bijection,forall suchx and Z. If h is Cartesian, the injectivity of h implies the cancellation property that h ∗ is “monic w.r. to p ”, meaning that for parallel arrows k,k′ in X with codomain X,and withp (k)=p (k′), wehavethat k.h=k′.h impliesk=k′. Forlateruse,we recall abasicfact: Lemma 1.1 If k=k′.h isCartesian,andh isCartesianthen k′ is Cartesian. 1 The functor p :X →B is called a fibrationif for every a :A→B in B and anyY ∈X withp (Y)=B,thereexistsaCartesianarrowovera withcodomainY. ThefibreoverA∈B isthecategorywhoseobjectsaretheX ∈X withp (X)=A, andwhosearrowsarearrowsinX whichbyp mapto1 ;sucharrowsarecalled A vertical(overA). All this is standard, dating back essentially to early French category theory (Grothendieck, Chevalley, Giraud, Bénabou,...). For a modern account, see [1] II.8.1, [2] B.1.3, or [3]. Note that these notions are elementary (they make sense for category objects in any left exact category), and they do not depend on the non-elementarynotionsofcleavage, orCat-valuedpseudofunctor. 2 The “factorization system” for a fibration Inthediagramsbelow,wetrytomakedisplayverticalarrowsvertically,andCarte- sianarrows horizontally. Recall from the literature that if p : X → B is a fibration, then every arrow z in X may be written as a compositeof a vertical arrow followedby a cartesian arrow. And, crucially, thisdecompositionof z is uniquemoduloa uniquevertical isomorphism. Or,equivalently,moduloanarrowwhichisatthesametimevertical and cartesian. (Recall that for vertical arrows, cartesian is equivalent to isomor- phism (= invertible).) This means that every arrow z in X may be represented by a pair (v,h) of arrows with v vertical and h cartesian, with z = v.h. Thus the codomainofvis thedomainofh. Wecall such apair a“vh compositionpair”, to maketheanalogywithvhspans,tobeconsideredbelow,moreexplicit. Twosuch pairs(v,h)and(v′,h′)representthesamearrowiffthereexistsaverticalcartesian (necessarily unique,and necessarily invertible)i suchthat v.i=v′ andi.h′ =h. (1) We say that (v,h) and (v′,h′) are equivalent if this holds. The composition of arrows in X can be described in terms of representative vh composition pairs, as follows. If z is represented by (v ,h ) for j = 1,2, then z .z is represented j j j 1 2 by (v .w,k.h ), where k is cartesian over p (h ) and w is vertical, and the square 1 2 1 2 displayedcommutes: · v 1 ❄ h · 1 ✲ · w v 2 ❄ ❄ · ✲ · ✲ · k h 2 Such k and w exists (uniquely, up to unique vertical cartesian arrows): construct first k as a cartesian lift of p (h ), then use the universal property of cartesian 1 arrowsto constructw. Thearrows z and z may be inserted, completingthe diagramwith two com- 1 2 mutative triangles, since z = v .h . But if we refrain from doing so, we have a j j j blueprint for a succinct and choice-free description of the fibrewise dual X ∗ of thefibration X →B. Note that a vh factorization of an arrow in X is much reminiscent of the factorizationforanE-M factorizationsystem,asin[Borceux]I.5.5,say,(withthe class of vertical arrows playing the role of E, and the class of cartesian arrows playingtheroleofM;however,notethatnoteveryisomorphisminX isvertical. 3 The dual fibration X ∗ The construction presented in this Section is still elementary, but requires more than just left exactness in the category where it is performed, namely exactness; this implies that good quotients exist for equivalence relations, and that maps on such a quotientcan be defined by assigningvalueson representativeelementsfor the equivalence classes. – We present the construction in the exact category of sets,forsimplicity. Given a fibration p :X →B. We describe another category X ∗ over B, as follows: The objects of X ∗ are the same as those of X ; the arrows X →Y are represented byvh spans,in thefollowingsense: 3 Definition3.1 Avh span inX fromX toY isa diagraminX of theform h · ✲ Y v (2) ❄ X withvvertical andh cartesian. The set of arrows in X ∗ from X to Y are equivalence classes of vh spans from X to Y, for the equivalence relation ≡ given by (v,h) ≡ (v′,h′) if there exists a verticalisomorphismi (necessarily unique)inX so that i.v.=v′ andi.h=h′. (3) We denote the equivalence class of the vh span (v,h) by {(v,h)}. They are the arrows of X ∗; the direction of a the arrow {(v,h)} is determined by its cartesian part h. Composition has to be described in terms of representative pairs; it is in fact the standard composite of spans, but let us be explicit: If z is represented by j (v ,h ) for j =1,2, then z .z is represented by (w,k), where k is cartesian over j j 1 2 p (h )and wis vertical,andthesquaredisplayedcommutes: 1 k h · ✲ · 2 ✲ · w v 2 ❄ ❄ · ✲ · (4) h 1 v 1 ❄ · Such k and w exists (uniquely, up to unique vertical cartesian arrows): construct first k as a cartesian lift of p (h ), then use the universal property of cartesian 1 4 arrowstoconstructw. (Thesquaredisplayedwillthenactuallybeapull-backdia- gram,thusthecompositiondescribedwillbethestandardcompositionofspans.) Compositionofvhspansdoesnotgiveadefinitevhspan,butratheran equiv- alence class of vh spans. So referring to (4), the compositeof {(v ,h )} with the 1 1 of{(v ,h )}is defined by 2 2 {(v ,h )}.{(v ,h )}:={(w.v ,k.h )}. 1 1 2 2 1 2 There is a functor p ∗ from X ∗ to B; on objects, it agrees with p :X →B; and p ∗({(v,h)}) = p (h). Note that if v : X′ → X is vertical, the vh span (v,1) represents amorphismX →X′ inX ∗. Clearly, a vertical arrow in X ∗ has a unique representative span of the form (v,1). So the fibres of p ∗ : X ∗ →B are canonically isomorphic to the duals of thefibres ofp :X →B, i.e. (X∗) ∼=(X )op;soX ∗ is“fibrewisedual”toX A A (butisnotingeneraldualtoX ,sincethefunctorp ∗:X ∗→Bisstillacovariant functor). The arrows in X ∗, we call comorphisms; it is ususally harmless to use thename“comorphism”alsoforarepresenting vhspan (v,h). There are twospecial classes ofcomorphisms: thefirst class consistsofthose comorphismsthat can berepresented by a pair(v,1)where 1 is the relevantiden- tity arrow. They are precisely the vertical arrows for X ∗ → B. – The second classconsistsofthosecomorphismsthatcanberepresentedbyapair(1,h)where 1 is the relevant identity arrow. We shall see that these are precisely the cartesian morphismsinX ∗. ’ Wefirst notethatif(v,h)represents an arbitrary arrow inX ∗,then (v,h)∈{(v,1)}.{(1,h)}; (5) thisiswitnessedbythediagram 1 h · ✲ · ✲ · 1 1 ❄ ❄ · ✲ · 1 v ❄ · 5 sincetheupperleft squareis oftheform consideredin (4). Proposition3.2 An arrow g is cartesian in X ∗ iff it admits a vh representative oftheform(1,h). Proof. Inonedirection,let(1,h)representacomorphismY →Z overb ∈B,and let (v,k) represent a comorphism X →Z over a .b . We display these data as the fullarrows inthefollowingdisplay(in X andB): · vX❄..............k...′...............✲..Yk ✲✲Z; h : : : · ✲ · ✲ · a b The dotted arrow k′ comes about by using the universal property of the cartesian arrow h in X . Since k and h are Cartesian, then so is k′, by the Lemma 1.1. So (v,k′)isacomorphismovera ,and(v,k′).(1.h)≡(v,k),andusingthecancellation property of Cartesian arrows, (v,k′) is easily seen to be the unique comorphism overa .b composingwith (1,h)togive(v,k). In the other direction, let g be a cartesian arrow in X ∗. Let (w,k) be an arbi- trary representative of g. Then by (5), g={(w,1)}.{(1,k)}. Since g is assumed cartesian in X∗, and {(1,k)} is cartesian by what is already proved, it follows from Lemma1.1 that {(w,1)}is cartesian. Since it is also vertical, it followsthat itisanisomorphisminX ∗,hencewisanisomorphisminX . Sincekiscartesian inX , w−1.k is cartesian as well,and (w,k)≡(1,w−1.k), sog has arepresentativeoftheclaimedform. Proposition3.3 Thefunctorp ∗ :X ∗ →B isa fibrationover B 6 Proof. Let b :A →B be an arrow in B, and letY be an object in over B. Since X →B is a fibration,there existsin X a cartesian arrow h overb , and then the vhspan (1,h)represents, by theabove,acartesian arrow inX ∗ overb . SinceX ∗ →B isafibration, wemayask foritsfibrewisedualX ∗∗: Proposition3.4 ThereisacanonicalisomorphismoverBbetweenX andX ∗∗. Proof. We describe an explicit functor y : X → X ∗∗. Let us denote arrows in X ∗ by dotted arrows; they may be presented by vh spans (v,h) in X . We first describe y on vertical and cartesian arrows separately. For a vertical v in X , say v : X′ → X, we have the vh span (v,1) in X , which represents a vertical arrow v:X′99KX inX ∗;thuswehaveavhspan(v,1)inX ∗,whichinturnrepresents a vertical arrow X → X′ in X ∗∗. This arrow, we take as y(v) ∈ X ∗∗. Briefly, y(v) =((v,1),1). – For a cartesian h : X′ →Y (over b , say), we have a vh span (1,h) in X , which represents a horizontal arrow h : X′ 99KY in X ∗ (cartesian overb ); thuswehaveavhspan(1,h)inX ∗,henceanarrow inX ∗∗, fromX′ to Y whichwetakeas y(h)∈X ∗∗;briefly, y(h)=(1,(1,v)). Then, for a general f : X →Y in X , we factor it v.h with v vertical and h cartesian, andputy(f):=y(v).y(h). Weleavetothereadertoverify thatadiffer- ent choice of v and h gives an equivalent vh span in X ∗, thus the same arrow in X ∗∗. Conversely, given an arrow g : X →Y in X ∗∗, represent it by a vh span in X ∗, (v,h), h X′ ...................✲.. Y . . . . . . . . . v. . . . . . . .❄ X Since v is vertical, we may pick a representative of v in the form (v,1) with v : X →X′, and since h is cartesian in X ∗, we may pick a representative of it if the form (1,h), with h:X′ →Y in X . Then the compositev.h:X →Y makes sense inX , and itgoes byyto thegiveng. Example. Consider a group homomorphism p : X → B. It is a fibration iff p is surjective. Assume this. Then the fibre (over the unique object ∗ of B) is the kernel K of p . Every h ∈ X is Cartesian; the vertical arrows are those of K . Then X∗ is canonically isomorphic to X . For, an element (arrow) (v,h) 7 of X ∗ may be presented by either (1,v−1.h), so may be presented in the form (1,x). The map (v,h) 7→ v−1.h gives a canonical isomorphism J : X ∗ → X . This isomorphism preserves p ; note that the p for X ∗ takes (v,h) to p (h). Let us for clarity denote it p ′, so p ′{(v,h)}=p (h). The kernel K ′ for p ′ consists of elementswhichmayberepresentedintheform(v,1)withv∈K ,soK ′may,asa set,beidentifiedwithK byidentifying(v,1)∈K ′⊆X ∗ withv∈K ⊆X . But this identification is an anti-isomorphism, since (v,1) by J goes to v−1.1 = v−1. So K ′ is identified as a group with K op. Thus we have a diagram of group homomorphisms (−)−1 K op ✲ K ∼ = i ⊆ ❄ J ❄ X ∗ ✲ X ∼ = p ′ p ❄ ❄ B ✲ B id wherei(v)={(v,1)}. IncasewhereB=1,andX isthegroupG,thefourmaps ofthetop squareare moreexplicitly‘thefourgroup isomorphisms (−)−1 Gop i ✲ G v7→{(v,1)} = ❄ J ❄ G∗ ✲ G {(v,h)}7→v−1.h where the inverse of J is given by h7→{(1,h)}. If we denote the inverse of J by j, wecan writetheinformationinthisdiagrammoresymmetrically: i j Gop ✲ G∗ ✛ G withi(v):={(v,1)}and j(h):={(1,h)}. 8 4 The case of a (pseudo-) functor Bop →Cat ItiswellknownthatapseudofunctorF :Bop→Cat,givesrisetoafibrationover B. Itisdescribedin,say,[2]B.1.3,orin[1]II.8.3. Thisfibrationisknownasthe GrothendieckconstructionforF. Wedescibeitbrieflyintermsofthefactorization systemalluded toinSection 1. Given a functor (or just a pseudo-functor) F : Bop →Cat. Then we have a category X whose objects are pairs (X,A) with A an object of B and X an object in F(A). Arrows (X,A) → (Y,B) are pairs (v,a ), where a : A → B and v:X →a ∗(Y)inF(A)(andwherea ∗ denotesthefunctorF(a ):F(B)→F(A)). Thefunctorp :X →B takes thisarrowto a . Let usdenotethearrow (1a ∗(Y),a )by a ⊳Y, thus a ⊳Y a ∗(Y) ✲ Y This is a Cartesian arrow over a in X , and every Cartesian arrow is of this form modulo unique vertical isomorphisms. There is then a canonical factorization of general arrows in X , namely, the arrow given by a pair (v,a ), as above, factors as (X,A) (v,1 ) . A ❄ (a ∗(Y),A) ✲ (Y,B) a ⊳Y LetF′beF followedbythedualizationfunctorCat→Cat. Thenamorphismover a inthefibrationcorrespondingtoF′,from(X,A)to(Y,B),isgivensimilarly,but nowwithv:a ∗(Y)→X,whichintermsofthecategoryF(A)ratherthan(F(A))op maybedisplayedinterms ofthevh span a ⊳Y (a ∗(Y),A) ✲ (Y,B) (v,1 ) , A ❄ (X,A) and from this, it is clear that the fibration corresponding to F′ is isomorphic to X ∗ as described inthepreviousSections. 9 One motivation for the present note is to extract the pure category theory behind “fibrewise contravariant functors" (like fibrewise duality for vector bundles), and “star- bundlefunctors”, asin[Kolaretal,1993]41.2. Thisisstillanongoing project. Icannotimaginethattheconstructions inthepresentnotearenotknown,butIdonot presently knowofanyavailable account. References [1] F.Borceux,HandbookofCategoricalAlgebra,vols.1-3,EncyclopediaofMathemat- icsanditsApplications 50-52(CambridgeUniversityPress1994). [2] P.T. Johnstone, Sketches of an Elephant: a Topos Theory Compendium, vols. 1-2, OxfordLogicGuides44and45(OxfordUniversityPress2002). [3] A.Kock,FibrationsasEilenberg-Moore Algebras, arXiv:1312.1608 v1,2013. [4] I. Kolar, P.W. Michor, and J. Slovak, Natural Operations in Differential Geometry, SpringerVerlag1993. UniversityofAarhus,January 2015 mail: kock(at)math.au.dk 10